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OF

G E 0 M E T RY.

FOR THE USE OF

SCHOOLS AND COLLEGES.

BY

CHARLES W. HACKLEY, S.T.D.,

PROFESSOR OF MATHEMATICS AND ASTRONOMY IN COLUMBIA COLLEGE,

AND AUTHOR OF A "TREATISE ON ALGEBRA,” ETC.

NEW YORK:

HARPER & BROTHERS, PUBLISHERS,

82 CLIFF STREET.

1 847.

Math 5082.

1892, Nov. 27, Gift of

ast,

Entered, according to Act of Congress, in the year one thousand

eight hundred and forty-seven, by

HARPER & BROTHERS,

in the Clerk's Office of the District Court of the Southern District

of New York.

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PREFACE.

The materials for the following work, like those of the author's treatise on Algebra, have been drawn from the latest and best foreign sources, and from the results of a varied experience of near twenty years as an instructor, commencing at the United States Military Academy.

The definitions of angles and parallel lines, upon which so much depends, will be found quite different from those in ordinary use; yet it is believed that no others hitherto suggested are so direct and distinct, so free from metaphysical objections, or so easily apprehended by the learner; and none, certainly, are productive of so much simplicity, generality, and brevity in the depending demonstrations.

The treatment of proportions as equalities of ratios, it is thought, will give greater clearness to the proof of those propositions in which they are used.

Great care has been taken to remedy all the little imperfections of demonstration in older treatises, and to supply some propositions which have been heretofore unaccountably omitted.

The infinitesimal system has been adopted without

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